Missing part of the proof of Master Theorem's case 2 (with ceilings and floors) in CLRS?
$begingroup$
I am trying to go through the proof of the Master Theorem in Introduction to Algorithms of Cormen, Leiserson, Rivest, Stein (CLRS). The theorem providers an asymptotic analysis for recurrence relations $T(n)=aT(n/b)+f(n)$ where $ageq 1, b > 1$ and $f(n)$ is an asymptotically positive function.
The authors create a formulation of the following lemma (4.3):
- if $f(n) = Theta(n^{log_ba})$, then $g(n)=Theta(n^{log_ba}{log_bn})$
where $g(n) = sum_{j=0}^{lfloor log_b n rfloor - 1} a^jf(n_j)$ and $n_j = begin{cases}
n, & text{if j = 0} \
lceil n_{j-1}/b rceil, & text{if j > 0}
end{cases}$
When it is proving for the situation in which floors and ceilings appear the autrors make a statement:
For case 2, we have $f(n)=Theta(n^{log_b a})$. If we can show that
$f(n_j) = O(n^{log_ba}/a^j) = O((n/b^j)^{log_ba})$, then the proof for
case 2 of Lemma 4.3 will go through.
Well, I got it. When this condition is true we can find some constant $c$, such that
$g(n) le csum_{j=0}^{lfloor log_b n rfloor - 1} n^{log_b a}$
and conclude that $g(n) = O(n^{log_b a}{log_bn})$.
But it's not a proof that $g(n) = Omega(n^{log_b a}{log_bn})$. Where is one? Should it be obvious? How can I prove this statement?
asymptotics
asked Dec 27 '18 at 20:18
MergasovMergasov
1104
$endgroup$
add a comment |
$begingroup$
I am trying to go through the proof of the Master Theorem in Introduction to Algorithms of Cormen, Leiserson, Rivest, Stein (CLRS). The theorem providers an asymptotic analysis for recurrence relations $T(n)=aT(n/b)+f(n)$ where $ageq 1, b > 1$ and $f(n)$ is an asymptotically positive function.
The authors create a formulation of the following lemma (4.3):
- if $f(n) = Theta(n^{log_ba})$, then $g(n)=Theta(n^{log_ba}{log_bn})$
where $g(n) = sum_{j=0}^{lfloor log_b n rfloor - 1} a^jf(n_j)$ and $n_j = begin{cases}
n, & text{if j = 0} \
lceil n_{j-1}/b rceil, & text{if j > 0}
end{cases}$
When it is proving for the situation in which floors and ceilings appear the autrors make a statement:
For case 2, we have $f(n)=Theta(n^{log_b a})$. If we can show that
$f(n_j) = O(n^{log_ba}/a^j) = O((n/b^j)^{log_ba})$, then the proof for
case 2 of Lemma 4.3 will go through.
Well, I got it. When this condition is true we can find some constant $c$, such that
$g(n) le csum_{j=0}^{lfloor log_b n rfloor - 1} n^{log_b a}$
and conclude that $g(n) = O(n^{log_b a}{log_bn})$.
But it's not a proof that $g(n) = Omega(n^{log_b a}{log_bn})$. Where is one? Should it be obvious? How can I prove this statement?
asymptotics
asked Dec 27 '18 at 20:18
MergasovMergasov
1104
$endgroup$
1
$begingroup$
"CLRS"? ${}{}{}$
$endgroup$
– Henning Makholm
Dec 27 '18 at 20:21
$begingroup$
@Henning Makholm It's an abbreviation from Cormen, Leiserson, Rivest, Stein, I refer to their book
$endgroup$
– Mergasov
Dec 27 '18 at 20:24
$begingroup$
I think there is something wrong with the equations. For instance what is $n_j$?
$endgroup$
– Mark
Dec 27 '18 at 20:31
$begingroup$
@Mark I have updated my question
$endgroup$
– Mergasov
Dec 27 '18 at 20:48
add a comment |
$begingroup$
I am trying to go through the proof of the Master Theorem in Introduction to Algorithms of Cormen, Leiserson, Rivest, Stein (CLRS). The theorem providers an asymptotic analysis for recurrence relations $T(n)=aT(n/b)+f(n)$ where $ageq 1, b > 1$ and $f(n)$ is an asymptotically positive function.
The authors create a formulation of the following lemma (4.3):
- if $f(n) = Theta(n^{log_ba})$, then $g(n)=Theta(n^{log_ba}{log_bn})$
where $g(n) = sum_{j=0}^{lfloor log_b n rfloor - 1} a^jf(n_j)$ and $n_j = begin{cases}
n, & text{if j = 0} \
lceil n_{j-1}/b rceil, & text{if j > 0}
end{cases}$
When it is proving for the situation in which floors and ceilings appear the autrors make a statement:
For case 2, we have $f(n)=Theta(n^{log_b a})$. If we can show that
$f(n_j) = O(n^{log_ba}/a^j) = O((n/b^j)^{log_ba})$, then the proof for
case 2 of Lemma 4.3 will go through.
Well, I got it. When this condition is true we can find some constant $c$, such that
$g(n) le csum_{j=0}^{lfloor log_b n rfloor - 1} n^{log_b a}$
and conclude that $g(n) = O(n^{log_b a}{log_bn})$.
But it's not a proof that $g(n) = Omega(n^{log_b a}{log_bn})$. Where is one? Should it be obvious? How can I prove this statement?
asymptotics
asked Dec 27 '18 at 20:18
MergasovMergasov
1104
$endgroup$
I am trying to go through the proof of the Master Theorem in Introduction to Algorithms of Cormen, Leiserson, Rivest, Stein (CLRS). The theorem providers an asymptotic analysis for recurrence relations $T(n)=aT(n/b)+f(n)$ where $ageq 1, b > 1$ and $f(n)$ is an asymptotically positive function.
The authors create a formulation of the following lemma (4.3):
- if $f(n) = Theta(n^{log_ba})$, then $g(n)=Theta(n^{log_ba}{log_bn})$
where $g(n) = sum_{j=0}^{lfloor log_b n rfloor - 1} a^jf(n_j)$ and $n_j = begin{cases}
n, & text{if j = 0} \
lceil n_{j-1}/b rceil, & text{if j > 0}
end{cases}$
When it is proving for the situation in which floors and ceilings appear the autrors make a statement:
For case 2, we have $f(n)=Theta(n^{log_b a})$. If we can show that
$f(n_j) = O(n^{log_ba}/a^j) = O((n/b^j)^{log_ba})$, then the proof for
case 2 of Lemma 4.3 will go through.
Well, I got it. When this condition is true we can find some constant $c$, such that
$g(n) le csum_{j=0}^{lfloor log_b n rfloor - 1} n^{log_b a}$
and conclude that $g(n) = O(n^{log_b a}{log_bn})$.
But it's not a proof that $g(n) = Omega(n^{log_b a}{log_bn})$. Where is one? Should it be obvious? How can I prove this statement?
asymptotics
asymptotics
asked Dec 27 '18 at 20:18
MergasovMergasov
1104
asked Dec 27 '18 at 20:18
MergasovMergasov
1104
edited Dec 27 '18 at 20:45
Mergasov
asked Dec 27 '18 at 20:18
MergasovMergasov
1104
asked Dec 27 '18 at 20:18
MergasovMergasov
1104
asked Dec 27 '18 at 20:18
MergasovMergasov
1104
1104
1
$begingroup$
"CLRS"? ${}{}{}$
$endgroup$
– Henning Makholm
Dec 27 '18 at 20:21
$begingroup$
@Henning Makholm It's an abbreviation from Cormen, Leiserson, Rivest, Stein, I refer to their book
$endgroup$
– Mergasov
Dec 27 '18 at 20:24
$begingroup$
I think there is something wrong with the equations. For instance what is $n_j$?
$endgroup$
– Mark
Dec 27 '18 at 20:31
$begingroup$
@Mark I have updated my question
$endgroup$
– Mergasov
Dec 27 '18 at 20:48
add a comment |
1
$begingroup$
"CLRS"? ${}{}{}$
$endgroup$
– Henning Makholm
Dec 27 '18 at 20:21
$begingroup$
@Henning Makholm It's an abbreviation from Cormen, Leiserson, Rivest, Stein, I refer to their book
$endgroup$
– Mergasov
Dec 27 '18 at 20:24
$begingroup$
I think there is something wrong with the equations. For instance what is $n_j$?
$endgroup$
– Mark
Dec 27 '18 at 20:31
$begingroup$
@Mark I have updated my question
$endgroup$
– Mergasov
Dec 27 '18 at 20:48
1
1
$begingroup$
"CLRS"? ${}{}{}$
$endgroup$
– Henning Makholm
Dec 27 '18 at 20:21
$begingroup$
"CLRS"? ${}{}{}$
$endgroup$
– Henning Makholm
Dec 27 '18 at 20:21
$begingroup$
@Henning Makholm It's an abbreviation from Cormen, Leiserson, Rivest, Stein, I refer to their book
$endgroup$
– Mergasov
Dec 27 '18 at 20:24
$begingroup$
@Henning Makholm It's an abbreviation from Cormen, Leiserson, Rivest, Stein, I refer to their book
$endgroup$
– Mergasov
Dec 27 '18 at 20:24
$begingroup$
I think there is something wrong with the equations. For instance what is $n_j$?
$endgroup$
– Mark
Dec 27 '18 at 20:31
$begingroup$
I think there is something wrong with the equations. For instance what is $n_j$?
$endgroup$
– Mark
Dec 27 '18 at 20:31
$begingroup$
@Mark I have updated my question
$endgroup$
– Mergasov
Dec 27 '18 at 20:48
$begingroup$
@Mark I have updated my question
$endgroup$
– Mergasov
Dec 27 '18 at 20:48
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3054323%2fmissing-part-of-the-proof-of-master-theorems-case-2-with-ceilings-and-floors%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3054323%2fmissing-part-of-the-proof-of-master-theorems-case-2-with-ceilings-and-floors%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
$begingroup$
"CLRS"? ${}{}{}$
$endgroup$
– Henning Makholm
Dec 27 '18 at 20:21
$begingroup$
@Henning Makholm It's an abbreviation from Cormen, Leiserson, Rivest, Stein, I refer to their book
$endgroup$
– Mergasov
Dec 27 '18 at 20:24
$begingroup$
I think there is something wrong with the equations. For instance what is $n_j$?
$endgroup$
– Mark
Dec 27 '18 at 20:31
$begingroup$
@Mark I have updated my question
$endgroup$
– Mergasov
Dec 27 '18 at 20:48